# On a new functional transform in analysis: The maximum transform

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### Cites background from "On a new functional transform in an..."

...Operations on fuzzy intervals lead to consider max-based "convolutions" of pairs of functions, which have been previously considered by Bellman and Karush (1961), in the framework of dynamic programming....

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### Cites background from "On a new functional transform in an..."

...Rearranging the terms in ( 5 ), the Lagrangian can be written as follows:...

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### "On a new functional transform in an..." refers background in this paper

...4) be defined by the equation (1) F(z) = max [e~f(x)]y z ^ O ....

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...Let (1) f(x) = max [fi(x1)f2(x2) • • • /N(XN)], R...

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...In the study of mathematical economics and operations research, we encounter the problem of determining the maximum of the function (1) F(xu x2, • • • , xN) = fi(xi) + f2(x2) + - - - +/N(XN) over the region R defined by xi+x2 + • • • +XN = X> x^O....

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...Differentiating this relation with respect to xy we have dz dz dz (1) F\z) — = (ƒ'(*) - zf(x))e~ - xf(x)e~ — = - xf(x)e~ — • ax dx dx Hence, (2) x = - F(z)/F(z), or F'(z) + xF(z) = 0....

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...Furthermore, since F(z) is unchanged when ƒ is replaced by its monotone envelope, we shall consider (1) only for monotone nondecreasing ƒ....

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